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24 votes
1 answer
1k views

About the abelian category of endofunctors of $\mathsf{Vect}$

Let $k$ be a field, $\mathsf{Vect}$ the category of finite dimensional vector spaces, and $\mathsf{C} = Fun(\mathsf{Vect},\mathsf{Vect})$ the abelian category of pointed endofunctors (sending $0$ to $...
Saal Hardali's user avatar
  • 7,789
15 votes
2 answers
869 views

What are the periodic Dyck paths?

I changed the thread completely so that everything is now elementary linear algebra. A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
Mare's user avatar
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11 votes
2 answers
558 views

Classification of algebras of finite global dimension via determinants of certain 0-1-matrices

I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background). A Morita-...
Mare's user avatar
  • 26.5k
7 votes
1 answer
462 views

On a problem for determinants associated to Cartan matrices of certain algebras

This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
Mare's user avatar
  • 26.5k
7 votes
0 answers
355 views

A homological algebra approach to the Union-closed sets conjecture

I noted a while ago that there is a nice homological formulation using incidence algebra of the Union-closed sets conjecture (https://en.wikipedia.org/wiki/Union-closed_sets_conjecture). It might just ...
Mare's user avatar
  • 26.5k
5 votes
0 answers
97 views

Periodics of Coxeter matrices for truncated Nakayama algebras

For $n \geq 3$ and $r \geq 3$ let $C_{n,r}=(c_{i,j})$ denote the $n \times n$-matrix where $c_{i,j}=1$ for $j=i,\dots,i+r-1$ (we only do this until $i+r-1>n$). So for example for $n=7$ and $r=3$ we ...
Mare's user avatar
  • 26.5k
4 votes
1 answer
101 views

Extension of scalars for bounded chain complexes of $kG$-modules

I'm wondering if a generalization regarding a statement from Curtis-Reiner holds. The original statement is as follows: (30.33) Theorem: Let $R$ and $S$ be complete discrete valuation rings, with $S$ ...
Sam K's user avatar
  • 175
4 votes
1 answer
423 views

Alternating sum of symmetric and exterior powers vanishes

Let $V$ be a vector space with some extra structure (maybe to be general, an object of an abelian tensor category?), where we can form tensor products and exterior powers $\Lambda^i V$ and symmetric ...
Kevin Casto's user avatar
  • 3,149
3 votes
0 answers
85 views

Exterior powers of the Cartan matrix and Dyck paths

(This question can be formulated purely combinatorially in terms of Dyck paths, which is done in the second part of the question. But I am more interested whether this can be explained by some sort of ...
Mare's user avatar
  • 26.5k
2 votes
0 answers
77 views

Equivalence of two descriptions of differentials of Koszul complex

My question comes from learning the paper [BGS96] Koszul duality patterns in representation theory by Beilinson, Ginzburg and Soergel, published in 1996. Let $A=T_{A_0}A_1/\langle R\rangle$ be a ...
L. Yhui's user avatar
  • 21